Energy level splitting for weakly interacting bosons in a harmonic trap

We consider identical quantum bosons with weak contact interactions in a two-dimensional isotropic harmonic trap. When the interactions are turned off, the energy levels are equidistant and highly degenerate. At linear order in the coupling parameter, these degenerate levels split, and we study the patterns of this splitting. It turns out that the problem is mathematically identical to diagonalizing the quantum resonant system of the two-dimensional Gross-Pitaevskii equation, whose classical counterpart has been previously studied in the mathematical literature on turbulence. Enhanced symmetry structures are known for this resonant system, and our purpose is to report on powerful implications these symmetries have for the level splitting in the quantum case. In particular, the symmetries relate energy shifts of different unperturbed levels so that infinitely many exactly integer energy differences survive at linear order in the interaction strength. The highest energy state emanating from each unperturbed level is explicitly described by our analytics. Considerable simplifications in computing the spectrum numerically result from exploiting the symmetries. We furthermore discuss the energy level spacing distributions in the spirit of quantum chaos theory. After separating the eigenvalues into blocks with respect to the known conservation laws, we observe the Wigner-Dyson statistics within such individual blocks, which suggests that no integrable structures beyond the symmetries we display should be expected in the problem.